The FMP Model

A general form of the FMP model is specified using the composite function,

\[ P(X_i = c | \theta) = \exp\left(\sum_{v=0}^c(b_{0i_{v}} + m_i(\theta))\right) / \left(\sum_{u=0}^{C_i - 1}\exp\left(\sum_{v=0}^u(b_{0i_{v}} + m_i(\theta))\right)\right) \]

where \(P\) indicates the probability of a response in category \(c\), \(c = 0, \ldots, C_i - 1\), \(\theta\) is the latent trait parameter, \(b_{iv}\) indicates an intercept for category \(v\), \(v = 1, \ldots, C_i - 1\), and \(\sum_{v=0}^0(b_{iv} + m_i(\theta)) \equiv 0\). In addition, let

\[ m_{i}(\theta)=b_{1i}\theta+b_{2i}\theta^{2}+\cdots+ b_{2k_{i}+1,i}\theta^{2k_{i}+1}, \] where \(2k_{i}+1\) equals the order of the polynomial for item \(i\), \(k_{i}\) is a nonnegative integer, and \(\boldsymbol{b}_{i}=(b_{0i_{1}},\ldots, b_{0i_{C_i-1}} b_{1i},\ldots,b_{2k_{i}+1,i})^{\prime}\) are item parameters that define the location and shape of the IRF. When \(k = 0\), the general FMP model reduces to the two-parameter item response model (for binary item responses) or the generalized partial credit model (for polytomous item responses).

For models with binary (0/1) item responses, flexmet also allows the use to include a lower asymptote parameter \(c_i\) and upper asymptote parameter \(d_i\) for the extended FMP model:

\[ P_i(\theta)=c_i + (d_i - c_i)[1+\exp(-m_{i}(\theta))]^{-1}. \]

The \(c_i\) and \(d_i\) parameters are unaffected by parameter transformations.

Transforming an Item Response Model

Below is a worked example of how to transform a two-parameter model to the expected sum score metric. The original two-parameter metric is denoted \(\theta\) and the expected sum score metric is denoted \(\theta^\star\). This example uses the 23 two-parameter model low self-esteem parameter estimates reported in Table 7 of Reise & Waller (2003).

First, we need to express the two-parameter model as an FMP model. The FMP model with \(k=0\) is identical to the slope-intercept parameterization of the two-parameter model. The Reise & Waller parameters are expressed on the more familiar difficulty-discrimination parameterization of the FMP model.

## example parameters from Table 7 of Reise & Waller (2003)
a <- c(0.57, 0.68, 0.76, 0.72, 0.69, 0.57, 0.53, 0.64,
       0.45, 1.01, 1.05, 0.50, 0.58, 0.58, 0.60, 0.59,
       1.03, 0.52, 0.59, 0.99, 0.95, 0.39, 0.50)
b <- c(0.87, 1.02, 0.87, 0.81, 0.75, -0.22, 0.14, 0.56,
       1.69, 0.37, 0.68, 0.56, 1.70, 1.20, 1.04, 1.69,
       0.76, 1.51, 1.89, 1.77, 0.39, 0.08, 2.02)

## convert from difficulties and discriminations to FMP parameters

b1 <- 1.702 * a
b0 <- - 1.702 * a * b
bmat <- cbind(b0, b1) 

The transformation from \(\theta\) to \(\theta^\star\) is defined by the test response function, which is the sum of item response functions:

\[ \theta^\star =\sum_iP_i(\theta) \]

There is usually not a closed form expression for \(\theta\) as a function of \(\theta^\star\). In addition, to transform the FMP item parmaeters, \(\theta\) must be expressed as a polynomial function of \(\theta^\star\):

\[ \theta = t_0 + t_1\theta^\star + t_2\theta^{\star2} + \cdots + t_{2k_\theta+1}\theta^{\star 2k_\theta+1}. \]

In this example, the metric transformation is known exactly, but it can be approximated by a monotonic polynomial. To approximate the metric transformation, we can generate a large number of observations and fit a monotonic polynomial function to the simulated values.

# generate a large number of theta and TRF (thetastar) values
theta <- seq(-3, 5, length = 5000)
TRF <- rowSums(irf_fmp(theta = theta, b = bmat))

Monotonic polynomial regression using the MonoPoly package can be used to approximate the metric transformation coefficients \(\boldsymbol{t}=(t_0,t_1,\ldots,t_{2k_\theta+1})^\prime\). We can fit a sequence of \(k_\theta\) values to find a good choice for the polynomial degree.

fmp0 <- MonoPoly::monpol(theta ~ TRF, K = 0)
fmp1 <- MonoPoly::monpol(theta ~ TRF, K = 1)
fmp2 <- MonoPoly::monpol(theta ~ TRF, K = 2)
fmp3 <- MonoPoly::monpol(theta ~ TRF, K = 3)
fmp4 <- MonoPoly::monpol(theta ~ TRF, K = 4)

Choose a “good enough” polynomial degree by looking at the residual sum of squares and by viewing patterns of residuals.

fmp0$RSS
#> [1] 59.88323
fmp1$RSS
#> [1] 7.231035
fmp2$RSS
#> [1] 7.225206
fmp3$RSS
#> [1] 0.2263184
fmp4$RSS
#> [1] 0.04808141

cols <- c("#E41A1C", "#377EB8", "#4DAF4A", "#984EA3", "#FF7F00")
par(lwd = 2)
curve(0*x, xlim = c(0, 22), ylim = c(-1, 1), col = "darkgray",
      xlab = "Expected Sum Score", 
      ylab = "Residuals of Polynomial Approximation")

points(TRF, residuals(fmp0), type = 'l', col = cols[1], lty = 2)
points(TRF, residuals(fmp1), type = 'l', col = cols[2], lty = 3)
points(TRF, residuals(fmp2), type = 'l', col = cols[3], lty = 2)
points(TRF, residuals(fmp3), type = 'l', col = cols[4], lty = 1)
points(TRF, residuals(fmp4), type = 'l', col = cols[5], lty = 3)

legend("bottomright",
       legend = c(expression(paste(italic(k[theta])," = 0")),
                  expression(paste(italic(k[theta])," = 1")),
                  expression(paste(italic(k[theta])," = 2")),
                  expression(paste(italic(k[theta])," = 3")),
                  expression(paste(italic(k[theta])," = 4"))),
       col = cols, lty = c(2, 3, 2, 1, 3), bty = "n")

Suppose we choose to retain the \(k_\theta = 3\) approximation. Then, the metric transformation vector equals,

(tvec <- coef(fmp3))
#>         beta0         beta1         beta2         beta3         beta4 
#> -3.807887e+00  2.141638e+00 -6.477732e-01  1.171824e-01 -1.208074e-02 
#>         beta5         beta6         beta7 
#>  7.022950e-04 -2.138094e-05  2.651772e-07

and the transformed item parameters equal

bstarmat <- t(apply(bmat, 1, transform_b, tvec = tvec))

## inspect transformed parameters
signif(head(bstarmat), 2)
#>      [,1] [,2]  [,3] [,4]   [,5]    [,6]     [,7]    [,8]
#> [1,] -4.5  2.1 -0.63 0.11 -0.012 0.00068 -2.1e-05 2.6e-07
#> [2,] -5.6  2.5 -0.75 0.14 -0.014 0.00081 -2.5e-05 3.1e-07
#> [3,] -6.1  2.8 -0.84 0.15 -0.016 0.00091 -2.8e-05 3.4e-07
#> [4,] -5.7  2.6 -0.79 0.14 -0.015 0.00086 -2.6e-05 3.2e-07
#> [5,] -5.4  2.5 -0.76 0.14 -0.014 0.00082 -2.5e-05 3.1e-07
#> [6,] -3.5  2.1 -0.63 0.11 -0.012 0.00068 -2.1e-05 2.6e-07

We can check that the transformation worked by plotting the test response function for the transformed model. If successful, this is a straight line because the latent trait \(\theta^\star\) should be as close as possible to the expected sum score.

par(pty = "s")
curve(rowSums(irf_fmp(x, bmat = bstarmat)), xlim = c(0, 23),
      ylim = c(0, 23), xlab = expression(paste(theta,"*")),
      ylab = "Expected Sum Score")
abline(0, 1, col = 2)

The bstarmat parameters can then be used as item parameters for subsequent analyses, such as trait score estimation.